NP-completeness and the real world. NP completeness. NP-completeness and the real world (2) NP-completeness and the real world

Transcription

1 -completeness and the real world completeness Course Discrete Biological Models (Modelli Biologici Discreti) Zsuzsanna Lipták Imagine you are working for a biotech company. One day your boss calls you and tells you that they have invented a new sequencing technology. It generates lots of fragments of the target molecule, which may overlap. Your job as chief algorithm designer is to write a program that reconstructs the target molecule. You get to work... Laurea Triennale in Bioinformatica a.a. 204/5, fall term 2/39 -completeness and the real world Imagine you are working for a biotech company. One day your boss calls you and tells you that they have invented a new sequencing technology. It generates lots of fragments of the target molecule, which may overlap. Your job as chief algorithm designer is to write a program that reconstructs the target molecule. You get to work... you find that... you need a superstring (of all fragments) ashortestsuperstring you need to maximize the overlaps any superstring is just a permutation of the fragments you need the/a best permutation (which maximizes total overlap) -completeness and the real world (2) But after weeks and weeks and weeks... all you have come up with is: Exhaustive algorithm List all possible permutations and choose the best. 2/39 3/39 -completeness and the real world (3) So you can go to your boss and say: This is no good, since for,000 fragments (typical experiment) this would need far longer than the age of the universe. Why? 000! permutations source: Wolfram Alpha say we need 000 operations per permutation (summing overlaps) if we have a computer that does billion (= 0 9 ) operations/sec it can handle million (0 6 ) permutations per second So it will take /0 6 = seconds Age of the universe: about 4 billion years seconds source: Wikipedia We can see that a million (0 6 ) or a billion (0 9 ) times faster computer wouldn t help much, either. Nor would faster handling of the permutations. Bad idea, you may get fired! source: Garey & Johnson, A Guide to the Theory of -completeness, 979 4/39 5/39

2 Or you could say: You prove that the reconstruction problem is -complete, and you say: Unfortunately, it is very hard to do impossibility proofs... source: Garey & Johnson, A Guide to the Theory of -completeness, 979 6/39 So it makes no sense to fire you and get another expert! source: Garey & Johnson, A Guide to the Theory of -completeness, 979 7/39 Overview Overview - class of that can be solved e ciently (i.e., in polynomial time) - class of that can solved in polynomial time by a non-deterministic Turing machine (nicer definition to follow) -complete -maximallydi cult in : every other problem can be transformed into these in polynomial time (details later) -hard - like -complete, but not necessarily in Usual way of drawing the classes and :!complete!hard 8/39 9/39 The class Definition A decision problem X is in if there is a polynomial time algorithm A which solves X. A decision problem is one that allows only YES or NO answers. s Given a sorted array of n numbers, is number x present? O(log n) time Given an array of n numbers, is number x present? O(n) time Given a graph G, isg Eulerian? O(n + m) time, where n = V, m = E(G). Details: Determine for each vertex whether it has even degree (or whether it is balanced if G is a digraph) in O(n) time;determinewithbfswhetherg is connected in O(n + m) time. 0 / 39 olynomial time algorithms Algorithm A is polynomial time if there is a polynomial p s.t. for every input I of size n, A terminates in at most p(n) steps. size is usually measured in bits, number of elements (for an integer array), number of vertices and edges (for graphs), number of characters (for strings) / 39

3 The class - class of that can solved in polynomial time by a non-deterministic Turing machine (non-deterministic polynomial time) alternative definition: = class of polynomial time checkable Whenever the answer is YES, there must exist a certificate (proof) of this, and it must be checkable (verifiable) in polynomial time. roblem: Given a graph G =(V, E), is G Hamiltonian (i.e. does it have a Hamiltonian cycle)? Certificate: A Hamiltonian cycle: check whether it is a cycle, and whether it contains every vertex exactly once, in O(n) time,wheren = V. 2 / 39 The class = class of polynomial time checkable Whenever the answer is YES, there must exist a certificate (proof) of this, and it must be checkable (verifiable) in polynomial time. roblem: Given a fragment set F of m fragments, does a superstring of length apple k exist? Certificate: asuperstrings of length apple k: checkforeveryf whether f is substring of S, ino( f + S ) time; altogether O( F + m S ) time,so polynomial in input size. input: F, inputsize: F = m i= fi, S apple i fi, som S apple F 2 3 / 39 The class and = class of polynomial time checkable Whenever the answer is YES, there must exist a certificate (proof) of this, and it must be checkable (verifiable) in polynomial time.!complete roblem: Given a complete digraph G =(V, E) with non-negative weights on edges, does it have a Hamiltonian path of weight at least r? Certificate: a Ham. path of weight r: check weight of path, check whether it contains every vertex exactly once, in O(n) time,wheren = V!hard 4 / 39 5 / 39 The = question The = question One of the big open question of computer science is: Is =? Since is clear, the question is:? Can every problem in be solved e ciently? I.e. is the shaded area empty?!complete!hard Most people believe: 6=. N.B.: There is a US $,000,000 prize for solving this question. 6 / 39 7 / 39

4 -complete and -hard -complete A problem X is -complete if it is in every problem in can be transformed/reduced to it in polynomial time (details soon) -hard A problem X is -hard if every problem in can be transformed/reduced to it in polynomial time (details soon) i.e., the only di erence is that it is not necessarily itself in. -complete and -hard Theorem Let X be an -complete or -hard problem. If we find a polynomial time algorithm for X,then =. Corollary Let X be an -complete or -hard problem. Then there is no polynomial time algorithm for X,unless =. Since we all believe that 6=, thismeansinpracticalterms: In practice Let X be an -complete or -hard problem. Then there is no polynomial time algorithm for X, fullstop. 8 / 39 9 / 39 -complete -complete Contains a list of known -complete : Michael R. Garey, David S. Johnson, Computers and Intractability - A Guide to the Theory of -completeness, 979 one of the best known and most cited books ever in computer science 20 / 39 2 / 39 -complete You have found your problem (the Shortest Common Superstring roblem) in the Garey-Johnson: Decision vs. optimization Optimization problem Given F, find a shortest common superstring S. Decision problem Given F, does a common superstring S exist with S apple k? N.B. If I can solve the decision problem, then I can also solve a reduced version of the optimization problem: determining the length of an SCS. Use log( F ) many calls to the decision problem. (Details: Determine the length of an SCS using binary search: Let N = F. We know that a common superstring exists with length N, namely the concatenation of all f 2 F. Nowsetk = N/2; if the answer is YES, continue with k = N/4, else with k =3N/4, etc.) 22 / / 39

5 Decision vs. optimization olynomial time reductions/transformations!complete!hard Now we want to talk about -complete. These are the ones to which all others in can be reduced in polynomial time. What does this mean? Many -hard are optimization versions of -complete. If the decision version is -complete, then the optimization version is automatically -hard. (Why?) So it s enough to prove that the decision version is -complete. (Or to find it in the Garey-Johnson.) Translate problem X to problem Y in polynomial time Definition We say that problem X is polynomial time reducible to problem Y, X apple p Y, if there is an algorithm A and a polynomial p, s.t.forevery instance I of X, A constructs an instance J of Y in time p( I ) such that: I is a YES-instance of X, J is a YES-instance of Y. 25 / / 39 An example reduction An example reduction SCS-decision Given: Fragment set F Question: Does a superstring of length apple k exist? {TACC, CGGACT, ACTAC, ACGGA} Weighted Hamiltonian path-decision Given: A weighted complete digraph G. Question: Does G have a Ham. path of weight F k? a = TACC c = CGGACT Let F contain m fragments of length r. Time for transformation: O(m 2 r 2 ), (or even faster): the time of the transformation is dominated by the time for computing the overlaps between the fragments, i.e. the weights on the edges This is polynomial in the input size F = mr (namely, with polynomial p(n) =n 2 ). YES on the left, YES on the right Therefore we have shown: SCS-decision apple p Weighted Hamiltonian path-decision b = ACTAC 2 d = ACGGA 27 / / 39 olynomial time reductions/transformations Definition (again) We say that problem X is polynomial time reducible to problem Y, X apple p Y, if there is an algorithm A and a polynomial p, s.t.forevery instance I of X, A constructs an instance J of Y in time p( I ) such that: N.B. I is a YES-instance of X, J is a YES-instance of Y. Think of X apple p Y as meaning: X is not harder than Y Note that we are still paying for the reduction/transformation: it takes polynomial time! olynomial time reductions/transformations -complete A problem X is -complete if it is in for every problem Y 2 : Y apple p X. Theorem Let X be an -complete problem. If we find a polynomial time algorithm for X,then =. There are important technical di erences between Karp-reductions/transformations (R. Karp, 972), and Turing-reductions (S. Cook, 97), which we are ignoring here. 29 / / 39

6 olynomial time reductions/transformations Is your problem -complete? Theorem Let X be an -complete problem. If we find a polynomial time algorithm for X,then =. roof Let A be a polytime algorithm for X.Wehavetoshow:. LetY be any problem in. We will now give a polytime algorithm for Y.LetI be an instance of Y.SinceX is -complete, there is an algorithm B which transforms any instance I of Y into an instance J of X in polynomial time, say in p B ( I ). In particular this implies J apple p B ( I ). The algorithm A for X solves J in time p A ( J ) apple p A (p B ( I )) (since all parameters are positive). Thus, we have an algorithm C = A B (B followed by A) fory, which gives an answer to instance I in time at most p C ( I ) :=p B ( I )+p A (p B ( I )), which, being a sum and composition of polynomials, is a polynomial in I. So if you have a computational problem, and you think it might be -complete, then Find it in the Garey-Johnson, or some other compendium of -complete/-hard, or prove that it is -complete. This is far beyond the scope of this course... You will learn how to do this in an advanced course on algorithms/complexity. 3 / / 39 Recap Why do we believe that 6=? : which are polynomial-time checkable (for YES-instances, a certificate exists which can be verified in polynomial time) polynomial time reduction/transformation, X apple p Y,means:instances of X can be transformed to instances of Y in polynomial time -complete : in to which all in can be polytime reduced -hard : like -complete but not necessarily in An e cient (=polytime) algorithm for an -complete or -hard problem would imply: =, which nobody believes is true Therefore, no e cient algorithm can exist for these (we all believe) Many, many important real-life are -complete or -hard. Finding an e cient algorithm for just one of these would prove =. Much much work has gone into finding e cient algorithms for these. By some of the best mathematicians and computer scientists on earth. No one has been able to find an e cient algorithm for any one of these so far. 33 / / 39 Why do we believe that 6=?... by some of the best mathematicians and computer scientists on earth... What to do next? If we find that our computational problem is -complete or -hard, then what can we do? despair/give up Run an exhaustive search algorithm This is often possible for small instances, often combined with specific tricks. Verify that your instances are really general. Often, the general case is -complete, but many special cases are not! E.g. the SBH-sequencing problem is not -complete: otherwise we could not have found a better formulation (Euler cycles in de Bruijn graphs) which is polynomially solvable! (continued on next page) 35 / / 39

7 What to do next? Summary (continued from previous page) Devise heuristics: algorithms that work well in practice but without guaranteeing the quality of the solution. olynomial time approximation algorithms: Algorithms that do not guarantee the optimal solution, but a solution which approximates the optimum. E.g. the Greedy Algorithm for SCS guarantees a solution which is at most 4 times longer than the optimum. and, and, and... There are certain for which no e cient algorithms exist (very, very, very, very probably) These are called -complete (decision ) or -hard (optimization ) Many real-life, also in computational biology/bioinformatics, are -complete/-hard There are ways of dealing with this (see previous list) 37 / / 39 Summary Merry Christmas! 39 / 39